reserve SAS for AffinPlane;

theorem Th3:
  SAS is translational implies for a,a9,b,b9,c,c9 being Element of
SAS holds ( not a,a9 // a,b & not a,a9 // a,c & a,a9 // b,b9 & a,a9 // c,c9 & a
  ,b // a9,b9 & a,c // a9,c9 implies b,c // b9,c9 )
proof
  assume
A1: SAS is translational;
  let a,a9,b,b9,c,c9 be Element of SAS such that
A2: not a,a9 // a,b and
A3: not a,a9 // a,c and
A4: a,a9 // b,b9 and
A5: a,a9 // c,c9 and
A6: a,b // a9,b9 & a,c // a9,c9;
  set A=Line(a,a9);
A7: a<>a9 by A2,AFF_1:3;
  then
A8: A is being_line by AFF_1:def 3;
  then consider C being Subset of SAS such that
A9: c in C and
A10: A // C by AFF_1:49;
A11: C is being_line by A10,AFF_1:36;
A12: a in A & a9 in A by AFF_1:15;
  then
A13: A<>C by A3,A8,A9,AFF_1:51;
A14: a,a9 // A by A8,A12,AFF_1:23;
  then a,a9 // C by A10,AFF_1:43;
  then c,c9 // C by A5,A7,AFF_1:32;
  then
A15: c9 in C by A9,A11,AFF_1:23;
  consider P being Subset of SAS such that
A16: b in P and
A17: A // P by A8,AFF_1:49;
A18: P is being_line by A17,AFF_1:36;
  a,a9 // P by A14,A17,AFF_1:43;
  then b,b9 // P by A4,A7,AFF_1:32;
  then
A19: b9 in P by A16,A18,AFF_1:23;
  A<>P by A2,A8,A12,A16,AFF_1:51;
  hence thesis by A1,A6,A8,A12,A16,A17,A9,A10,A18,A11,A19,A15,A13,AFF_2:def 11;
end;
